Thai Journal of Mathematics
Volume 8 (2010) Number 2 : 405–417
www.math.science.cmu.ac.th/thaijournal
Online ISSN 1686-0209
Existence and Iterative Approximation of
a Unique Solution of a System of General
Quasi-Variational Inequality Problems
K.R. Kazmi1 , F.A. Khan2 and Mohd. Shahzad
Abstract : This paper deals with the existence and uniqueness of solution of a
system of general quasi-variational inequality problems in a uniformly smooth Banach space. Further, a Mann-type partial implicit iterative algorithm is proposed
for the system of general quasi-variational inequality problems and discussed its
convergence analysis. The iterative algorithms and results presented here improve
the iterative algorithms and results discussed in [3,7,8].
Keywords : System of general quasi-variational inequality problems, Mann-type
partial implicit iterative algorithms, strongly accretive mapping, Lipschitz continuous mapping, relaxed cocoercive mapping.
2000 Mathematics Subject Classification : 49J40, 47J20, 65K10.
1
Introduction
In recent years, the theory of variational inequalities has become an effective and powerful tool for studying a wide range of problems arising in many
diverse fields of pure and applied sciences. One of the most interesting and important problems in the theory of variational inequalities is the development of
an efficient iterative algorithm to compute approximate solutions of variational
inequality problems.
One of the efficient numerical techniques for solving variational inequality
1
Corresponding author; E-mail: [email protected]; This work has been done under
a Major Research Project No. F.36-7/2008 (SR) sanctioned by the University Grants
Commission, Government of India, New Delhi
2
Supported by NBHM, Department of Atomic Energy, Government of India under
Grants-in-aid for Post-doctoral fellowship (Reference no. 40/2/2007-R&D II/3910)
c 2010 by the Mathematical Association of Thailand.
Copyright reserved.
All rights
406
Thai J. Math. 8(2) (2010)/ K.R. Kazmi et al.
problems in Hilbert spaces is the projection method and its variant forms. Since
the standard projection method strictly defined on the inner product property of
Hilbert spaces, it can no longer be applied for variational inequality problems in
Banach spaces. This fact motivates us to develop alternative method to study iterative algorithms for approximating the solutions of variational inequality problems
in Banach spaces.
In 2004, Verma [8] studied the convergence analysis of a iterative algorithm
for approximating the solution of system of variational inequality problems involving relaxed cocoercive mappings in Hilbert spaces. Since then, many authors
developed and studied different iterative methods for various classes of variational
inequality problems and systems of variational inequality problems involving relaxed cocoercive mappings in Hilbert spaces, see for example [3-7].
We remark that the conditions used in the main results of [3-8] reduced the
relaxed cocoercive mappings into strongly monotone mappings and thus the results
are actually for variational inequality problems for strongly monotone mappings,
see Lemma 2.5.
In this paper, we consider a system of general quasi-variational inequality
problems (in short, SGQVIP) in a uniformly smooth Banach space. Further,
using retraction method, we prove the existence of unique solution for SGQVIP.
Furthermore, a Mann-type partial implicit iterative algorithm is given for SGQVIP
and discussed its convergence analysis.
2
Preliminaries
We assume that E is a real Banach space equipped with norm k · k; E ∗ is the
topological dual space of E; 2E is the power set of E; h·, ·i is the dual pair between
∗
E and E ∗ and J : E → 2E is the normalized duality mapping defined by
J(x) = {f ∈ E ∗ : hx, f i = kxk2 , kxk = kf k}, x ∈ E.
First, we recall and define the following concepts and results which are needed
in the sequel.
Definition 2.1[1]. A Banach space E is called smooth if, for every x ∈ E with
kxk = 1, there exists a unique f ∈ E ∗ such that kf k = f (x) = 1. The modulus of
smoothness of E is the function ρE : [0, ∞) → [0, ∞) defined by
ρE (τ ) = sup
n (kx + yk + kx − yk)
2
o
− 1 : x, y ∈ E, kxk = 1, kyk = τ .
Definition 2.2[10]. The Banach space E is said to be uniformly smooth if
lim
τ →0
ρE (τ )
= 0.
τ
407
Existence and iterative approximation of a unique solution. . .
Throughout the rest of the paper, we assume that E is a real uniformly smooth
Banach space. We observe that if E is smooth then J is single-valued and if E = H,
a Hilbert space then J is the identity map on H.
Lemma 2.1[2]. Let J : E → E ∗ be the normalized duality mapping. Then for
all x, y ∈ E, we have
(a) kx + yk2 ≤ kxk2 + 2hy, J(x + y)i;
4kx − yk p
(b) hx − y, J(x) − J(y)i ≤ 2d2 ρE
, where d = (kxk2 + kyk2 )/2.
d
Definition 2.3[2]. Let K be a nonempty, closed and convex subset of E. A
mapping PK : E → K is said to be
2
(i) retraction if PK
= PK ;
(ii) nonexpansive retraction if kPK (x) − PK (y)k ≤ kx − yk, ∀x, y ∈ E;
(iii) sunny retraction if PK (PK (x) − t(x − PK (x)) = PK (x), ∀x ∈ E, t ∈ R+ .
Lemma 2.2[2]. A retraction PK is sunny and nonexpansive if and only if
hx − PK (x), J(PK (x) − y)i ≥ 0, ∀x, y ∈ E.
Definition 2.3. A mapping T : E → E is said to be:
(i) β-Lipschitz continuous if there exists a constant β > 0 such that
kT x − T yk ≤ βkx − yk,
∀x, y ∈ E;
(ii) r-strongly accretive if there exists a constant r > 0 such that
hT x − T y, J(x − y)i ≥ rkx − yk2 ,
∀x, y ∈ E;
(iii) r-relaxed accretive if there exists a constant r > 0 such that
hT x − T y, J(x − y)i ≥ − rkx − yk2 ,
∀x, y ∈ E;
(iv) α-cocoercive (or α-inverse strongly accretive) if there exists a constant α > 0
such that
hT x − T y, J(x − y)i ≥ αkT x − T yk2,
∀x, y ∈ E;
(v) relaxed (γ, r)-cocoercive if there exist constants γ, r > 0 such that
hT x − T y, J(x − y)i ≥ − γkT x − T yk2 + rkx − yk2 ,
(vi) λ-expansive if there exists a constant λ > 0 such that
kT x − T yk ≥ λkx − yk,
∀x, y ∈ E.
∀x, y ∈ E;
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Thai J. Math. 8(2) (2010)/ K.R. Kazmi et al.
Remark 2.1. We observe that a r-strongly accretive mapping must be a relaxed
(γ, r)-cocoercive mapping, or a γ-cocoercive mapping must be a relaxed (γ, r)cocoercive mapping whenever r = 0, but the converse is not true, see [8].
Lemma 2.3[9]. Suppose {δn }∞
n=0 is a nonnegative sequence satisfying the following inequality
δn+1 ≤ (1 − an )δn + σn , ∀ n ≥ 0,
∞
P
with an ∈ [0, 1],
an = ∞, and σn = o(an ). Then lim δn = 0.
n→∞
n=0
Next, we prove the following results.
Lemma 2.4. If the mapping T : E → E is relaxed (γ, r)-cocoercive then T is
2r − 1 1/2
1
-expansive for r > .
1 + 2γ
2
Proof. Since T is relaxed (γ, r)-cocoercive mapping then there exist constants
r, γ > 0 such that
−γkT x − T yk2 + rkx − yk2 ≤ hT x − T y, J(x − y)i
≤ kT x − T yk kx − yk
1
≤
kT x − T yk2 + kx − yk2 .
2
Hence, we have
(1 + 2γ)kT x − T yk2 ≥ (2r − 1)kx − yk2 ,
or
kT x − T yk ≥
2r − 1 1/2
1 + 2γ
kx − yk, for r >
1
.
2
This completes the proof.
Lemma 2.5. Let the mapping T : E → E be relaxed (γ, r)-cocoercive and βLipschitz continuous.
(i) If γβ 2 < r then T is (r − γβ 2 )-strongly accretive.
(ii) If r < γβ 2 then T is (r − γβ 2 )-relaxed accretive.
Proof. Since T is relaxed (γ, r)-cocoercive and β-Lipschitz continuous, we have
hT x − T y, J(x − y)i ≥ − γkT x − T yk2 + rkx − yk2
≥ − γβ 2 kx − yk2 + rkx − yk2
= (r − γβ 2 )kx − yk2 .
This completes the proof.
409
Existence and iterative approximation of a unique solution. . .
Remark 2.2.(i) In the spirit of Lemma 2.5(i), Theorem 3.1 [5], Theorem 3.1 [6,7],
Theorem 2.1 [8], Theorem 2.1 [4] and Theorem 3.1 [3] are actually for variational
inequality problems for strongly monotone mappings in a√
Hilbert space.
1 4rγ − 1
1
(ii) It is easily observed from Lemma 2.5 that β − ≥
and γr > .
γ
2γ
4
3
System of general quasi-variational inequality
problems
Let K : E×E → 2E be a set-valued mapping such that for each (x, y) ∈ E×E,
K(x, y) be a nonempty, closed and convex set in E. Let g : E → E be a singlevalued mapping and F, G : E × E → E be nonlinear mappings. We consider the
problem of finding x∗ , y ∗ ∈ E with g(x∗ ), g(y ∗ ) ∈ K(x∗ , y ∗ ) ∩ K(y ∗ , x∗ ) 6= ∅ such
that
hρ1 F (y ∗ , x∗ ) + g(x∗ ) − g(y ∗ ), J(z1 − g(x∗ ))i ≥ 0, ∀ z1 ∈ K(y ∗ , x∗ ),
hρ2 G(x∗ , y ∗ ) + g(y ∗ ) − g(x∗ ), J(z2 − g(y ∗ ))i ≥ 0, ∀ z2 ∈ K(x∗ , y ∗ ),
(3.1)
which we call a system of general quasi-variational inequality problems (SGQVIP).
Special cases:
(1) If g ≡ I, identity mapping and E = H, Hilbert space, then SGQVIP
(3.1) reduces to a system of quasi-variational inequality problems of finding
x∗ , y ∗ ∈ K(x∗ , y ∗ ) ∩ K(y ∗ , x∗ )(6= ∅) such that
hρ1 F (y ∗ , x∗ ) + x∗ − y ∗ , z1 − x∗ i ≥
hρ2 G(x∗ , y ∗ ) + y ∗ − x∗ , z2 − y ∗ i ≥
0,
0,
∀z1 ∈ K(y ∗ , x∗ ),
∀z2 ∈ K(x∗ , y ∗ ),
which is the correct form of the system (1)-(2) studied by Noor and Huang
[7]. We remark that to find the solution (x∗ , y ∗ ) of system (1)-(2) [7] is not
equivalent to find (x∗ , y ∗ ) such that
x∗ = PK(y∗ ,x∗ ) [y ∗ − ρ1 F (y ∗ , x∗ )]
y ∗ = PK(x∗ ,y∗ ) [x∗ − ρ2 G(x∗ , y ∗ )],
unless x∗ , y ∗ ∈ K(y ∗ , x∗ ) ∩ K(x∗ , y ∗ )(6= ∅).
(2) If F = G = T , SGQVIP (3.1) is equivalent to finding x∗ , y ∗ ∈ E with
g(x∗ ), g(y ∗ ) ∈ K(x∗ , y ∗ ) ∩ K(y ∗ , x∗ )(6= ∅) such that
hρ1 T (y ∗ , x∗ ) + g(x∗ ) − g(y ∗ ), J(z1 − g(x∗ ))i
hρ1 T (x∗ , y ∗ ) + g(y ∗ ) − g(x∗ ), J(z2 − g(y ∗ ))i
which appears to be new one.
≥ 0,
≥ 0,
∀z1 ∈ K(y ∗ , x∗ ),
∀z2 ∈ K(x∗ , y ∗ ),
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Thai J. Math. 8(2) (2010)/ K.R. Kazmi et al.
(3) If K(x∗ , y ∗ ) = K(y ∗ , x∗ ) = K, a nonempty, closed and convex set in E,
then SGQVIP (3.1) reduces to the following system of variational inequality
problems of finding x∗ , y ∗ ∈ K such that
hρ1 F (y ∗ , x∗ ) + g(x∗ ) − g(y ∗ ), J(z − g(x∗ ))i
hρ2 G(x∗ , y ∗ ) + g(y ∗ ) − g(x∗ ), J(z − g(y ∗ ))i
≥ 0,
≥ 0,
∀z ∈ K,
∀z ∈ K.
For appropriate and suitable choice of operators F, G and the set-valued mapping K, one can obtain a number of new and previously known problems from the
SGQVIP (3.1) as special cases.
Next, we have the following Lemma.
Lemma 3.1. (x∗ , y ∗ ) ∈ E × E with g(x∗ ), g(y ∗ ) ∈ K(x∗ , y ∗ ) ∩ K(y ∗ , x∗ ), is a
solution of SGQVIP (3.1) if and only if (x∗ , y ∗ ) satisfies
g(x∗ ) = PK(y∗ ,x∗ ) [g(y ∗ ) − ρ1 F (y ∗ , x∗ )],
g(y ∗ ) = PK(x∗ ,y∗ ) [g(x∗ ) − ρ2 G(x∗ , y ∗ )],
where ρ1 , ρ2 > 0 are constants.
4
Existence and uniqueness of solution for SGQVIP
(3.1)
First, we define the following concepts.
Definition 4.1. Let g : E → E be a nonlinear mapping. A mapping F : E × E →
E is said to be:
(i) α-strongly accretive with respect to g in the first argument if there exists a
constant α > 0 such that
hF (x1 , y1 )−F (x2 , y2 ), J(g(x1 )−g(x2 ))i ≥ αkx1 −x2 k2 ,
∀ x1 , x2 , y1 , y2 ∈ E.
(ii) β-Lipschitz continuous in the first argument if there exists a constant β > 0
such that
kF (x1 , y1 ) − F (x2 , y2 )k ≤ βkx1 − x2 k,
∀x1 , x2 , y1 , y2 ∈ E.
We remark that if g is µ-Lipschitz continuous, then α ≤ µβ.
Now, we prove the existence and uniqueness of solution for SGQVIP(3.1).
Theorem 4.1. Let E be a uniformly smooth Banach space with ρE (t) ≤ ct2 for
some c > 0; K be a nonempty, closed and convex set in E and let the mapping
411
Existence and iterative approximation of a unique solution. . .
g : E → E be σ-strongly accretive and µ-Lipschitz continuous. Let F, G : K ×K →
E be two mappings such that F be α1 -strongly accretive with respect to g and
β1 -Lipschitz continuous in the first argument and G be α2 -strongly accretive with
respect to g and β2 -Lipschitz continuous in the first argument. Suppose that there
is a constant λ > 0 such that
kPK(x1 ,y1 ) (x) − PK(x2 ,y2 ) (x)k ≤ λkx1 − x2 k,
(4.1)
∀(x1 , y1 ), (x2 , y2 ) ∈ E × E, x ∈ E and ρ1 , ρ2 > 0 satisfy the following conditions:
p
θ + θ2 < 1 and θ + θ1 < 1,
(4.2)
p
p
where θ := (1 − 2σ + 64cµ2 ); θ1 := (µ2 − 2ρ1 α1 + 64cρ21 β12 ) + λ;
q
θ2 := (µ2 − 2ρ2 α2 + 64cρ22 β22 ) + λ.
Then SGQVIP (3.1) has a unique solution.
Proof. For (x, y) ∈ E × E, define a mapping Q : E × E → E × E by
∀(x, y) ∈ E × E,
(4.3)
T (x, y) = x − g(x) + PK(y,x) [g(y) − ρ1 F (y, x)],
(4.4)
S(x, y) = y − g(y) + PK(x,y) [g(x) − ρ2 G(x, y)],
(4.5)
Q(x, y) = (T (x, y), S(x, y)),
where T, S : E × E → E are defined by
and
where ρ1 , ρ2 > 0 are some constants.
For any (x1 , y1 ), (x2 , y2 ) ∈ E × E, it follows that from (4.1) and (4.4) that
≤
≤
≤
kT (x1 , y1 ) − T (x2 , y2 )k
kx1 − x2 − (g(x1 ) − g(x2 ))k
+kPK(y1 ,x1 ) [g(y1 ) − ρ1 F (y1 , x1 )] − PK(y2 ,x2 ) [g(y2 ) − ρ1 F (y2 , x2 )]k
kx1 − x2 − (g(x1 ) − g(x2 ))k
+kPK(y1 ,x1 ) [g(y1 ) − ρ1 F (y1 , x1 )] − PK(y2 ,x2 ) [g(y1 ) − ρ1 F (y1 , x1 )]k
+kPK(y2 ,x2 ) [g(y1 ) − ρ1 F (y1 , x1 )] − PK(y2 ,x2 ) [g(y2 ) − ρ1 F (y2 , x2 )]k
kx1 − x2 − (g(x1 ) − g(x2 ))k + λky1 − y2 k
+kg(y1 ) − g(y2 ) − ρ1 (F (y1 , x1 ) − F (y2 , x2 ))k.
(4.6)
Similarly,
kS(x1 , y1 ) − S(x2 , y2 )k
≤ ky1 − y2 − (g(y1 ) − g(y2 ))k + λkx1 − x2 k
+kg(x1 ) − g(x2 ) − ρ2 (G(x1 , y1 ) − G(x2 , y2 ))k.
(4.7)
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Thai J. Math. 8(2) (2010)/ K.R. Kazmi et al.
Since g is σ-strongly accretive and µ-Lipschitz continuous, by using Lemma
2.1, we have
≤
=
≤
=
kx1 − x2 − (g(x1 ) − g(x2 ))k2
kx1 − x2 k2 − 2hg(x1 ) − g(x2 ), J(x1 − x2 − g(x1 ) − g(x2 )))i
kx1 − x2 k2 − 2hg(x1 ) − g(x2 ), J(x1 − x2 )i
−2hg(x1 ) − g(x2 ), J(x1 − x2 − (g(x1 ) − g(x2 ))) − J(x1 − x2 )i
kx1 − x2 k2 − 2σkx1 − x2 k2 + 64cµ2 kx1 − x2 k2
(1 − 2σ + 64cµ2 )kx1 − x2 k2 .
(4.8)
Similarly, we have
ky1 − y2 − (g(y1 ) − g(y2 ))k2 ≤ (1 − 2σ + 64cµ2 )ky1 − y2 k2 .
(4.9)
Since F is α1 -strongly accretive with respect to g in the first argument and
β1 -Lipschitz continuous in the first argument, and G is α2 -strongly accretive with
respect to g in the first argument and β2 -Lipschitz continuous in the first argument,
we have
≤
≤
=
kg(y1 ) − g(y2 ) − ρ1 (F (y1 , x1 ) − F (y2 , x2 ))k2
kg(y1 ) − g(y2 )k2 − 2ρ1 hF (y1 , x1 ) − F (y2 , x2 ), J(g(y1 ) − g(y2 )i
−2ρ1 hF (y1 , x1 ) − F (y2 , x2 ), J(g(y1 ) − g(y2 ) − ρ1 (F (y1 , x1 ) − F (y2 , x2 )))
−J(g(y1 ) − g(y2 ))i
µ2 ky1 − y2 k2 − 2ρ1 α1 ky1 − y2 k2 + 64cρ21 β12 ky1 − y2 k2
(µ2 − 2ρ1 α1 + 64cρ21 β12 )ky1 − y2 k2 ,
which implies
≤
kg(y1 ) − g(y2 ) − ρ1 (F (y1 , x1 ) − F (y2 , x2 ))k
q
(µ2 − 2ρ1 α1 + 64cρ21 β12 )ky1 − y2 k.
(4.10)
Similarly, we have
≤
kg(x1 ) − g(x2 ) − ρ2 (G(x1 , y1 ) − G(x2 , y2 ))k
q
(µ2 − 2ρ2 α2 + 64cρ22 β22 )kx1 − x2 k.
(4.11)
From (4.6), (4.8) and (4.10), we have
kT (x1 , y1 )−T (x2 , y2 )k
≤
q
p
(1 − 2σ + 64cµ2 )kx1 −x2 k+
(µ2 − 2ρ1 α1 + 64cρ21 β12 )+λ ky1 −y2 k. (4.12)
Also, from (4.7), (4.9) and (4.11), we have
kS(x1 , y1 )−S(x2 , y2 )k
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Existence and iterative approximation of a unique solution. . .
≤
q
p
(1 − 2σ + 64cµ2 )ky1 −y2 k+
(µ2 − 2ρ2 α2 + 64cρ22 β22 )+λ kx1 −x2 k. (4.13)
From (4.12) and (4.13), we have
kT (x1 , y1 )−T (x2 , y2 )k+kS(x1 , y1 )−S(x2 , y2 )k
where
≤ k1 kx1 − x2 k + k2 ky1 − y2 k
≤ max k1 , k2 kx1 − x2 k + ky1 − y2 k ,
(4.14)
k1 := θ + θ2 and k2 := θ + θ1 .
(4.15)
Now, define the norm k · k∗ on E × E by
k(x, y)k∗ = kxk + kyk, ∀(x, y) ∈ E × E.
(4.16)
We can easily observe that (E × E, k · k∗ ) is a Banach space. Hence, it follows
from (4.3), (4.14) and (4.16) that
kQ(x1 , y1 ) − Q(x2 , y2 )k∗ ≤ max{k1 , k2 }k(x1 , y1 ) − (x2 , y2 )k∗ .
(4.17)
Since k1 < 1, k2 < 1 by condition (4.2), it follows from (4.17) that Q is a
contraction mapping. Hence, by the Banach contraction principle, there exists a
unique (x, y) ∈ E × E such that Q(x, y) = (x, y), which implies that
g(x) = PK(y,x) [g(y) − ρ1 F (y, x)],
g(y) = PK(x,y) [g(x) − ρ2 G(x, y)].
It follows from Lemma 3.1, that (x, y) is the unique solution of SGQVIP (3.1).
This completes the proof.
5
Iterative algorithms and convergence analysis
In this section, we suggest that the fixed-point formulation for SGQVIP(3.1),
see Lemma 3.1, and Theorem 4.1 are very important from the numerical approximation point of view and help to suggest the following iterative algorithm for
SGQVIP(3.1).
Mann-type partially implicit iterative algorithm (in short, MTPIIA)
5.1. For a given point (x0 , y0 ) ∈ E × E, compute an approximate solution (xn , yn )
given by iterative schemes:
xn+1 = (1−an )xn +an [xn −g(xn )+PK(yn ,xn ) (g(yn )−ρ1 F (yn , xn ))],
(5.1)
yn+1 = (1−bn )xn+1 +bn [yn+1 −g(yn+1 )+PK(xn+1 ,yn ) (g(xn+1 )−ρ2 G(xn+1 , yn ))],
(5.2)
∞
P
where ρ1 , ρ2 > 0 are constants and an ,bn ∈ (0, 1], n ≥ 0 with
an = ∞ and
lim bn =1.
n→∞
n=0
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Thai J. Math. 8(2) (2010)/ K.R. Kazmi et al.
Special cases:
(1) If E ≡ H and g ≡ I, MTPIIA 5.1 reduces to the following iterative algorithm. For a given point (x0 , y0 ) ∈ H × H compute an approximate solution
(xn , yn ) given by
xn+1 = (1 − an )xn + an PK(yn ,xn ) (yn − ρ1 F (yn , xn )),
yn+1 = (1 − bn )xn+1 + bn PK(xn+1 ,yn ) (xn+1 − ρ2 G(xn+1 , yn )),
which is mainly due to Noor and Huang [7].
(2) If K(x, y) ≡ K(y, x) ≡ K, the nonempty, closed and convex set in E and
g ≡ I, MTPIIA 5.1 reduces to the following iterative algorithm. For a given
point (x0 , y0 ) ∈ E × E, compute an approximate solution (xn , yn ) given by
xn+1 = (1 − an )xn + an PK (yn − ρ1 F (yn , xn )),
yn+1 = (1 − bn )xn+1 + bn PK (xn+1 − ρ2 G(xn+1 , yn )),
where an , bn ∈ [0, 1] ∀n ≥ 0.
(3) If an ≡ bn ≡ 1; K(y, x) = K(x, y) ≡ K and g ≡ I, MTPIIA 5.1 reduces
to the following iterative algorithm. For a given point (x0 , y0 ) ∈ E × E,
compute an approximate solution (xn , yn ) given by
xn+1 = PK (yn − ρ1 F (yn , xn )),
yn+1 = PK (xn+1 − ρ2 G(xn+1 , yn )).
Finally, we discuss the convergence criteria for MTPIIA 5.1.
Theorem 5.1. Let the mappings E, F, G, g be same as in Theorem 4.1 and
let conditions (4.1) and (4.2) of Theorem 4.1 hold. Then approximate solution
(xn , yn ) generated by MTPIIA 5.1, converges strongly to the unique solution (x, y)
of SGQVIP(3.1).
Proof. It follows from Theorem 4.1 that SGQVIP (3.1) has the unique solution
(x, y) ∈ E × E. Hence, by Lemma 3.1, we have
x = (1 − an )x + an [x − g(x) + PK(y,x) (g(y) − ρ1 F (y, x))],
(5.3)
y = (1 − bn )y + bn [y − g(y) + PK(x,y) (g(x) − ρ2 G(x, y))].
(5.4)
From (4.1),(5.1) and (5.3), we have
kxn+1 −xk ≤ (1−an )kxn −xk+an kxn −x−(g(xn )−g(x))k
+an kPK(yn ,xn ) (g(yn ) − ρ1 F (yn , xn )) − PK(y,x) (g(y) − ρ1 F (y, x))k
Existence and iterative approximation of a unique solution. . .
415
≤ (1 − an )kxn − xk + an kxn − x − (g(xn ) − g(x))k
+an kPK(y,x) (g(yn ) − ρ1 F (yn , xn )) − PK(y,x) (g(y) − ρ1 F (y, x))k
+an kPK(yn ,xn ) (g(yn ) − ρ1 F (yn , xn )) − PK(y,x) (g(yn ) − ρ1 F (yn , xn ))k
≤ (1 − an )kxn − xk + an kxn − x − (g(xn ) − g(x))k
+an kg(yn ) − g(y) − ρ1 (F (yn , xn ) − F (y, x))k + an λkyn − yk.
(5.5)
Since g is σ-strongly accretive and µ-Lipschitz continuous, using Lemma 2.1,
we have
kxn −x−(g(xn )−g(x))k2 ≤ (1−2σ+64cµ2)kxn −xk2 .
(5.6)
Also, F is α1 -strongly accretive with respect to g in the first argument and
β1 -Lipschitz continuous in the first argument, we have
kg(yn )−g(y)−ρ1(F (yn , xn )−F (y, x))k2 ≤ (µ2 −2ρ1 α1 +64cρ21β12 )kyn −yk2.
(5.7)
From (5.5),(5.6) and (5.7), it follows that
kxn+1 −xk
p
≤ (1−an )kxn −xk+an (1 − 2σ + 64cµ2 )kxn −xk
q
+an (µ2 − 2ρ1 α1 + 64c1 ρ21 β12 )kyn −yk+anλkyn −yk
q
p
= (1−an [1− (1 − 2σ + 64cµ2 )])kxn −xk+an
(µ2 − 2ρ1 α1 + 64cρ21 β12 )+λ kyn −yk
= (1−an (1−θ))kxn −xk+an θ1 kyn −yk.
(5.8)
Now, from (4.1), (5.2) and (5.4), we have
kyn+1 −yk ≤ (1−bn )kxn+1 −xk+(1−bn)ky −xk+bnkyn+1 −y −(g(yn+1)−g(y))k
+bn kg(xn+1 ) − g(x) − ρ2 (G(xn+1 , yn ) − G(x, y))k + bn λkxn+1 − xk
(5.9)
Since g is σ-strongly accretive and µ-Lipschitz continuous, using Lemma 2.1,
we have
kyn −y −(g(yn )−g(y))k2 ≤ (1−2σ+64cµ2 )kyn −yk2 .
(5.10)
Since G is α2 -strongly accretive with respect to g in the first argument and
β2 -Lipschitz continuous in the first argument, we have
kg(xn+1 )−g(x)−ρ2 (G(xn+1 , yn )−G(x, y))k2 ≤ (µ2 −2ρ2 α2 +64cρ22β22 )kxn+1 −xk2 .
(5.11)
From (5.6), (5.9), (5.10) and (5.11), it follows that
416
Thai J. Math. 8(2) (2010)/ K.R. Kazmi et al.
kyn+1 −yk
p
≤ (1−bn )kxn+1 −xk+(1−bn)ky−xk+bn (1 − 2σ + 64cµ2 )kyn+1 −yk
q
+bn (µ2 − 2ρ2 α2 + 64cρ22 β22 )kxn+1 −xk+bn λkxn+1 −xk
q
= 1−bn 1−( (µ2 − 2ρ2 α2 + 64cρ22 β22 )+λ) kxn+1 −xk
p
+bn (1 − 2σ + 64cµ2 )kyn+1 −yk+(1−bn)ky−xk
= (1−bn (1−θ2 ))kxn+1 −xk+bn θkyn+1 −yk+(1−bn)ky−xk
≤ kxn+1 − xk + θkyn+1 − yk + (1 − bn )ky − xk,
p
where θ2 := (µ2 − 2ρ2 α2 + 64cρ22 β22 ) + λ < 1; bn ∈ (0, 1].
(5.12)
From (5.12), we have
kyn+1 − yk ≤
1 kxn+1 − xk + (1 − bn )ky − xk .
(1 − θ)
(5.13)
Combining (5.8) and (5.13), we have
kxn+1 −xk ≤ (1−an (1−θ))kxn −xk+an θ1
=
n
o
1 kxn −xk+(1−bn−1)ky−xk
(1 − θ)
h
h
θ1 ii
θ1 (1 − bn−1 )
1 − an 1 − θ +
kxn − xk + an
ky − xk.
1−θ
1−θ
(5.14)
∞
P
θ1 By condition (4.2) it follows that 1 − θ +
∈ (0, 1],
an 1 − θ +
1−θ
n=0
θ1 an θ1 (1 − bn−1 )
= o(an ), and then by Lemma 3.2, it follows
= ∞, and
1−θ
(1 − θ)
that lim kxn − xk = 0. Further, the result lim kyn − yk = 0 follows from (5.13)
n→∞
n→∞
and lim bn = 1. This completes the proof.
n→∞
Remark 5.1. The proof of theorems presented in this paper for SGQVIP(3.1)
under the assumption of relaxed cocoercivity on mappings F and G need further
research effort.
Acknowledgement : The authors express their sincere thanks to the referee for
his/her valuable comments and suggestions for improving the first version of this
paper.
Existence and iterative approximation of a unique solution. . .
417
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(Received 18 August 2009)
K.R. Kazmi, F.A. Khan
Department of Mathematics,
Aligarh Muslim University,
Aligarh 202002 India.
e-mail : [email protected] (K.R. Kazmi)
Mohd. Shahzad
Department of Applied Mathematics,
Aligarh Muslim University,
Aligarh 202002 India.